Question

use dimensional analysis to solve the following problems: 60. determine the number of years in 8.35 x 10^6 minutes. 61. sixty - two months is equivalent to how many seconds? a gas station is charging $1.299 per gallon of gas. what would be the price for a liter of gas? 63. what portion of a year is 3.97 x 10^15 nanoseconds?

Explanation:

Step1: Recall time - unit conversion factors

1 minute = 60 seconds, 1 hour = 60 minutes, 1 day = 24 hours, 1 year = 365 days

Step2: Convert 8.35×10⁶ minutes to years

First, convert minutes to hours: $8.35\times 10^{6}\text{ minutes}\times\frac{1\text{ hour}}{60\text{ minutes}}=\frac{8.35\times 10^{6}}{60}\text{ hours}$
Then convert hours to days: $\frac{8.35\times 10^{6}}{60}\text{ hours}\times\frac{1\text{ day}}{24\text{ hours}}=\frac{8.35\times 10^{6}}{60\times24}\text{ days}$
Finally, convert days to years: $\frac{8.35\times 10^{6}}{60\times24}\text{ days}\times\frac{1\text{ year}}{365\text{ days}}$
\[

$$\begin{align*} &=\frac{8.35\times 10^{6}}{60\times24\times365}\text{ years}\\ &=\frac{8.35\times 10^{6}}{525600}\text{ years}\\ &\approx 15.9\text{ years} \end{align*}$$

\]

Step3: Convert 62 months to seconds

1 year = 12 months, 1 day = 24 hours, 1 hour = 60 minutes, 1 minute = 60 seconds
First, convert months to years: $62\text{ months}\times\frac{1\text{ year}}{12\text{ months}}=\frac{62}{12}\text{ years}$
Then convert years to days: $\frac{62}{12}\text{ years}\times365\text{ days/year}=\frac{62\times365}{12}\text{ days}$
Then convert days to hours: $\frac{62\times365}{12}\text{ days}\times24\text{ hours/day}=\frac{62\times365\times24}{12}\text{ hours}$
Then convert hours to minutes: $\frac{62\times365\times24}{12}\text{ hours}\times60\text{ minutes/hour}=\frac{62\times365\times24\times60}{12}\text{ minutes}$
Finally, convert minutes to seconds: $\frac{62\times365\times24\times60}{12}\text{ minutes}\times60\text{ seconds/minute}= 160776000\text{ seconds}$

Step4: Convert price per gallon to price per liter

1 gallon = 3.78541 liters
If the price is $1.299 per gallon, then the price per liter is $1.299\div3.78541\approx 0.343$ dollars per liter

Step5: Convert 3.97×10¹⁵ nanoseconds to years

1 second = 10⁹ nanoseconds, 1 minute = 60 seconds, 1 hour = 60 minutes, 1 day = 24 hours, 1 year = 365 days
First, convert nanoseconds to seconds: $3.97\times 10^{15}\text{ nanoseconds}\times\frac{1\text{ second}}{10^{9}\text{ nanoseconds}} = 3.97\times10^{6}\text{ seconds}$
Then convert seconds to minutes: $3.97\times 10^{6}\text{ seconds}\times\frac{1\text{ minute}}{60\text{ seconds}}=\frac{3.97\times 10^{6}}{60}\text{ minutes}$
Then convert minutes to hours: $\frac{3.97\times 10^{6}}{60}\text{ minutes}\times\frac{1\text{ hour}}{60\text{ minutes}}=\frac{3.97\times 10^{6}}{60\times60}\text{ hours}$
Then convert hours to days: $\frac{3.97\times 10^{6}}{60\times60}\text{ hours}\times\frac{1\text{ day}}{24\text{ hours}}=\frac{3.97\times 10^{6}}{60\times60\times24}\text{ days}$
Finally, convert days to years: $\frac{3.97\times 10^{6}}{60\times60\times24}\text{ days}\times\frac{1\text{ year}}{365\text{ days}}$
\[

$$\begin{align*} &=\frac{3.97\times 10^{6}}{60\times60\times24\times365}\text{ years}\\ &=\frac{3.97\times 10^{6}}{15768000}\text{ years}\\ &\approx 0.25\text{ years} \end{align*}$$

\]

Answer:

1. Approximately 15.9 years
2. 160776000 seconds
3. Approximately $0.343 per liter
4. Approximately 0.25 years